If you would like to make yourself a public match go to DNA Relatives and click on "Update Your Profile" located on the far right side of the top row. Click on the drop-down next to "Profile Name" and select "Show me as Your Name." Click on "Save Changes" in the bottom left.
By making your profile name visible in DNA Relatives, you'll make your profile name visible in your public profile, if it's not already, and in the Countries of Ancestry tool. Your name and profile information will also be searchable by other 23andMe users.
Your profile picture will be shown to your matches in DNA Relatives if you choose to make your name or your community nickname visible.